Deterministic Chaos

6.H. Mathematical Definition of Chaos

Devaney, a function f:R -> R exhibits deterministic chaos if it satisfies three properties:

sensitivity to initial conditions arbitrarily close to every point x, there is a point y with fⁿ(x) and fⁿ(y) iterating far apart.

dense periodic points arbitrarily close to every point x, there is a point y with f^m(y) = y for some m.

mixing for every pair of intervals I and J, for some k f^k(J) and I overlap.

Click each name to see a graphical examples of that characteristics.

The bad news and the good news of chaos.

The first technical use of the term chaos.

Recent updates on the three conditions defining chaos.

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sensitivity to initial conditions	arbitrarily close to every point x, there is a point y with fⁿ(x) and fⁿ(y) iterating far apart.
dense periodic points	arbitrarily close to every point x, there is a point y with f^m(y) = y for some m.
mixing	for every pair of intervals I and J, for some k f^k(J) and I overlap.